© 2008 pearson addison wesley. all rights reserved chapter fourteen game theory

© 2008 Pearson Addison Wesley. All rights reserved

Chapter Fourteen

Game Theory

© 2008 Pearson Addison Wesley. All rights reserved. 14-2

Game Theory

• In this chapter, we examine three main topics

– An Overview of Game Theory– Static Games– Dynamic Games


An Overview of Game Theory

• Game theory– A set of tools that economists, political

scientist, political scientists, military analysts, and others use to analyze decision-making by players that use strategies



• Game– Any competition between players (firms) in

which strategic behavior plays a major role• Strategy

– A battle plan of the actions a player (firm) plans to take to compete with other players (firms)



• Strategic Behavior

— A set of actions a firm takes to increase

its profit, taking into account the possible

actions of other firms



• Game theory tries to answer two questions: how to describe a game and how to predict the game’s outcome.

• A game is described in terms of the players; its rules; the outcome; the payoffs to players corresponding to each possible outcome; and the information that players have about their rivals’ moves.



• The rules of the game determine the timing of players’ moves and the actions that players can make at each move.

• Games with complete information are games where the payoff function is common knowledge among all players.

• Games with perfect information are games where the player who is about to move knows the full history of the play of the game to this point, and that information is updated with each subsequent action.


• A static game is one in which each player acts only once and the players act simultaneously.

• A dynamic game is one where players move either sequentially or repeatedly.

• Sequential move dynamic games: chess, Stackelberg model.



Static Games

• A static game is one in which the players choose their actions simultaneously, have complete information about the payoff function, and play the game once.

• Examples: the Cournot and Bertrand models.


Normal-Form Games

• A normal-form representation of a static game of complete information specifies the player in the game, their possible strategies, and the payoffs for each combination of strategies.

• A profit matrix (or payoff matrix), such as in Table 14.1, shows the strategies the firms may choose and the resulting profits.


Table 14.1Profit Matrix for a Quantity-Setting Game: Dominant Strategy


Normal-Form Games

• Because the firms choose their strategies simultaneously, each firm selects a strategy that maximizes its profit given what it believes the other firm will do.

→a noncooperative game of imperfect information


Predicting A Game’s Outcome

• Dominant strategy– A strategy that strictly dominates (gives

higher profits than) all other strategies, regardless of the actions chosen by rival firms

• Where a firm has a dominant strategy, its belief about its rivals’ behavior is irrelevant.



• If United choose the high-output strategy ( ), American’s high-output strategy maximizes its profit.

• If United chooses the low-output strategy ( ), American’s high-output strategy maximizes its profit.

• Thus the high-output strategy is American’s dominant strategy.

48Uq

64Uq



• They don’t cooperate due to a lack of trust: Each firm uses the low-output strategy only if the firms have a binding agreement.



• Each firm has a substantial profit incentive to cheat on the agreement.

• In this type of game—called a prisoner’s dilemma game—all players have dominant strategies that lead to a profit (or other payoff) that is inferior to what they could achieve if they cooperated and pursued alternative strategies.


Table 14.2Profit Matrix for a Quantity-Setting Game: Iterated Dominance


Best Response and Nash Equilibrium

• We can think of oligopolies as engaging in a game, which is any competition between players (such as firms) in which strategic behavior plays a major role.



• When is an oligopolistic market in equilibrium?

• Nash equilibrium– A set of strategies such that, holding the

strategies of all other firms constant, no firm can obtain a higher profit by choosing a different strategy



• In a Nash equilibrium. No firm wants to change its strategy because each firm is using its best response —the strategy that maximizes its profit, given its beliefs about its rivals’ strategies.


Table 14.3Simultaneous Entry Game


Multiple Nash Equilibria

• Pure Strategy: Each player chooses a single action.

• Mixed Strategy: The player chooses among possible actions according to probabilities it assigns.


Multiple Nash Equilibria

• This game has two Nash equilibria in pure strategies: Firm 1 enters and Firm 2 does not, or Firm 2 enters and Firm 1 does not.

• How do the players know which (if any) Nash equilibrium will result? They don’t. It is difficult to see how the firms choose strategies unless they collude.


Mixed Strategies

• These pure Nash equilibria are unappealing because they call for identical firms to use different strategies. The firms may use the same strategies if their strategies are mixed. When both firms enter with a probability of one-half—say , if a flipped coin comes up heads—there is a Nash equilibrium in mixed strategies.


Mixed Strategies

• One important reason for introducing the concept of a mixed strategy is that some games have no pure-strategy Nash equilibria.

• In the game with no dominant strategies, neither firm has a strong reason to believe that the other will choose a pure strategy. It may think about its rival’s behavior as random.


Table 14.4(a)Advertising Game

Cooperation

• The game in panel (a) is a prisoners’ dilemma game. Each firm has a dominant strategy: to advertise.

• The sum of the firms’ profits is not maximized in this simultaneous-choice one-period game.


Cooperation

• Suppose the two firms meet in advance and agree not to advertise. Each firm has a substantial profit incentive to cheat on the agreement.

• This is because that in this game, if one firm advertises, its sales increase so that its profit rises, but its rival loses customers and hence the rival’s profit falls.



Table 14.4(b)Advertising Game

Cooperation

• In panel (b), when either firm advertises, the promotion attracts new customers to both firms. Advertising is a dominant strategy for both firms.

• Both firms advertises in the games in panel (a) and panel (b). The distinction is that the Nash equilibrium in which both advertise is the same as the collusive equilibrium in panel (b), but not in panel (a).


Dynamic Games

• In dynamic games players move sequentially or move simultaneously repeatedly over time, so a player has perfect information about other players’ previous moves.

• Extensive form of a game specifies the n players, the sequence in which they make their moves, the actions they can take at each move, the information that each player has about players’ previous moves, and the payoff function over all possible strategies.



Sequential Game - Stackelberg Model

• Suppose, however, that one of the firms, called the leader, can set its output before its rival, the follower, sets its output.

• This type of game, in which the players make decisions sequentially, arises naturally if one firm enters a market before another.


Figure 14.1 Stackelberg Game Tree

• The game tree shows the order of the firms move, each firm’s possible strategies at the time of its move, and the resulting profits.


Figure 14.1Stackelberg Game Tree


Stackelberg Game Tree

• If American choose 48, United will set 64, so American’s profit will be $3.8 million.




Stackelberg Game Tree

• Thus to maximize its profit, American choose 96. United responds by selling 48.

• This outcome is a Stackelberg (Nash) equilibrium.

Subgame Perfect Nash Equiligrium

• A set of strategies forms a subgame perfect Nsh equilibrium if the players’ strategies are a Nash equilibrium in every subgame.

• We can solve for the subgame perfect Nash equilibrium using backward induction, where we first determine the best response by the last player to move, next determine the best response for the player who made the next-to-last move, and then repeat the process until we reach the move at the beginning of the game.



Figure 14.2 Whether an Incumbent Pays to Prevent Entry

• If the potential entrant stays out of the market, it makes no profit, , and the incumbent firm makes the monopoly profit, . If the potential entrant enters the market, both firms make the duopoly profit, . Entry occurs if the duopoly profit is positive, .

0 e

i m

0 d

d


Figure 14.2 Whether an Incumbent Pays to Prevent Entry

• Entry is blockaded (does not occur regardless of actions by the incumbent) if the duopoly profit is negative because of low demand or high fixed costs of entering, both of which lower profit, if entry is not blockaded, the incumbent acts to deter entry by paying for exclusive rights to be the only firm at the rest stop only if . Otherwise (if ), the incumbent accommodates entry.

m db m db


Figure 14.2Game Tree: Whether an Incumbent Pays to Prevent Entry


Credibility

• credible threat– an announcement that a firm will use a

strategy harmful to its rival that the rivals believe because the firm’s strategy is rational in the sense that it is in the firm’s best interest to use it


Credibility

• By committing to produce a large quantity whether or not entry occurs, the incumbent discourages entry.

• Commitment as a Credible Threat. The intuition for why commitment makes a threat credible is that of “burning bridges.”

• Similarly, by limiting its future options, a firm makes itself stronger.


Repeated Game

• In a single-period game, one firm cannot punish the other firm for cheating in a cartel agreement.

• But if the firms meet period after period, a wayward firm can be punished by the other.


Repeated Game

• Supergame– A game that is played repeatedly, allowing

players to devise strategies for one period that depend on rivals’ actions in previous periods


Repeated Game

• In a repeated game, a firm can influence its rival’s behavior by signaling and threatening to punish.

• AA may signal UA by producing low output.• In addition to or instead of signaling, a firm

can punish a rival for not restricting output.


Repeated Game

• American will produce the smaller quantity each period as long as United does the same.

• If United produces the larger quantity in period , American will produce the larger quantity in period and all subsequent periods.

1tt


Repeated Game

• Thus United’s best policy is to produce the lower quantity in each period unless it cares greatly about current profit and little about future profits.

• Thus if firms play the same game indefinitely, they should find it easier to collude.


Repeated Game

• Maintaining a cartel will be difficult if the game has a known stopping point.

• If the players know that the game will end but aren’t sure when, cheating is less likely to occur.

• Collusion is therefore more likely in a game that will continue forever or that will end at an uncertain time.

© 2008 pearson addison wesley. all rights reserved chapter fourteen game theory

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